# Midterm [Practical Part]

* We want to approximate this data using linear regressions and Fuzzy c means


## Create a set of clusters from the data using FCM.

* Extend your 1d implementation of clustering to Nd
* After fit, for each point you will have
* N_CLUSTERS probas
* argmax(N_CLUSTERS) - corresponded cluster

* I used N_CLUSTERS = 3:

## For each cluster, fit linear regression locally

### Normal equation


* For each cluster we fit linreg - [theta0, theta1]
* Finally we have N_CLUSTERS linregs, shape is (N_CLUSTERS, 2)

### Draw this regressions:


## Create final predictor (degranularized model)


* Ai(xj) - proba that xj belongs to the cluster i
* ai - coefficients of i-th linear regression
* shape(A) is (N_POINTS, N_CLUSTERS)
* shape(a) is (N_CLUSTERS, 2)
* shape(X) is (N_POINTS, 2) // second dimension is for bias

## This formula can be vectorized

y = (X @ a.T * A).sum(1)

* @ is matrix multiplication
* "*" is elementwise multiplication
* .sum(1) summation over first(second) dimension
* shapes: ((N_POINTS, 2) @ (2, N_CLUSTERS) "*" (N_POINTS, N_CLUSTERS)).sum(1) -> (N_POINTS, 1)
### So for any xi we have weighted sum of linear regressions outputs where each weight is probability that this point belongs to this cluster.

## Calculate mse


* Implement mean squared from scratch or use sklearn

## Plot degranularized model